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Transmission Performance Analysis of RV Reducers Influenced by Profile applied sciences Article Transmission Performance Analysis of RV Reducers Influenced by Profile Modification and Load Hui Wang, Zhao-Yao Shi *, Bo Yu and Hang Xu Beijing Engineering Research Center of Precision Measurement Technology and Instruments, College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing 100124, China; [email protected] (H.W.); [email protected] (B.Y.); [email protected] (H.X.) * Correspondence: [email protected]; Tel.: +86-139-1030-7299 Received: 31 August 2019; Accepted: 25 September 2019; Published: 1 October 2019 Featured Application: A new multi-tooth contact model proposed in this study provides an effective method to determine the optimal profile modification curves to improve transmission performance of RV reducers. Abstract: RV reducers contain multi-tooth contact characteristics, with high-impact resistance and a small backlash, and are widely used in precision transmissions, such as robot joints. The main parameters affecting the transmission performance include torsional stiffness and transmission errors (TEs). However, a cycloid tooth profile modification has a significant influence on the transmission accuracy and torsional stiffness of an RV reducer. It is important to study the multi-tooth contact characteristics caused by modifying the cycloid profile. The contact force is calculated using a single contact stiffness, inevitably affecting the accuracy of the result. Thus, a new multi-tooth contact model and a TE model of an RV reducer are proposed by dividing the contact area into several differential elements. A comparison of the contact force obtained using the finite element method and the test results of an RV reducer prototype validates the proposed models. On this basis, the influence of load on the different modification methods is studied, including a TE, the mechanical performance, and the transmission efficiency. In addition, the proposed reverse profile is particularly suitable for situations with a large clearance and torque. This study provides a reliable theoretical basis for a multi-tooth contact analysis of a cycloid profile modification. Keywords: RV reducer; contact dynamics; transmission errors; cycloid mechanism 1. Introduction RV reducers are planocentric reduction mechanisms used for precise transmission applications. Such mechanisms incorporate numerous cycloid-pin teeth that are engaged concurrently. Therefore, RV reducers have a compact and highly rigid construction that is robust against shock-load and overloading. Furthermore, a minimal backlash and inertia assure a rapid acceleration and extremely accurate positioning. An RV reducer is ideally suited for precision mechanical control in industrial robots, machine tools, and assembly equipment, where precise positioning and a torsional stiffness capability are required [1]. Transmission errors (TEs) are key parameters used to represent the transmission performance of an RV reducer. The TE measurement method of a precision planetary cycloid reducer is defined in GB/T 37718-2019 [2]. The optimal measurement speed can be determined according to the Stribeck friction model [3,4]. The determination of optimal measurement speed of transmission errors could improve the measurement precision. A cycloid profile modification is one of the factors causing the TEs of an RV reducer [5–7]. However, to avoid the influence of machining errors on an RV reducer assemble, a cycloid Appl. Sci. 2019, 9, 4099; doi:10.3390/app9194099 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 4099 2 of 19 tooth profile modification is commonly used in engineering practice. The modification methods often include a modification of the moved distance, equidistance, rotation angle, and their combination [8]. An angular modification needs to be used with other methods concurrently. In [9], based on a tooth contact analysis, the TE caused by different combinations of modifications is quantitatively analyzed. To reduce the impact of a modification on the RV reducer transmission accuracy, the optimal cycloidal tooth profile modification method has been studied by numerous researchers. In [10], a modification method for designing the modification gap curves by adjusting the positions of five key points on the cycloid profile was proposed. This method can improve the carrying capacity of cycloidal drive, eliminate noise and vibration and develop the transmission accuracy. Compared with the designed modified tooth profile, the actual tooth profile with machining errors can be accurately measured to compensate such errors between the actual and designed tooth profiles [11,12]. In addition, judging the optimal modified tooth profile according to the amplitude frequency vibration response of the transmission system caused by different modified tooth profiles is an effective method [13]. The torsional stiffness of an RV reducer achieves a relatively large benefit from its multi-tooth contact characteristics, and thus can ensure a high transmission accuracy. Therefore, the contact force calculation of a cycloid-pin is particularly important when multiple teeth mesh together to transmit a power load [14,15]. Among the many studies conducted in this area, Xu proposed a numerical method to determine the number of teeth that are loaded simultaneously [16,17]. The model can be used to investigate the effect of changing the geometrical design parameters of the tooth profile of the cycloid gear. However, the contact area of each mesh tooth is treated as a whole, and the contact force is calculated using a single contact stiffness. Such a treatment method results in some errors in the actual contact status [18]. To reduce this deficiency, Ma established a hybrid contact force model. The contact cylinder surface is divided into many differential elements using the discrete element theory. Each differential element calculates the contact force separately according to the compression depth. Therefore, the contact model is more accurate for a complex profile contact [19]. Studies on cycloid tooth profile modification have mostly focused on the design method of optimizing the tooth profile, and the evaluation index is often limited to the transmission accuracy. The influences of the modification of the mechanical properties, contact area, and transmission efficiency have rarely been reported. However, such performances are also of concern among users of an RV reducer. In addition, the most important point that is often neglected is the load impact on the tooth profile modification. Regarding the multi-tooth contact model introduced in the references herein, the accuracy of the results will be significantly affected because it is based on a single contact stiffness model. In this paper, each tooth contact area is replaced by multiple differential contact units, and a multi-tooth contact model and a TE model of an RV reducer are first established. The proposed models are then verified using the finite element method and experimental measurements. On this basis, the influence of load on the different modification methods is studied in detail. The evaluation index considered includes the transmission accuracy, mechanical performance, and transmission efficiency. Finally, considering the load influence, the proper applications of different modification methods are discussed. 2. Analytical Model of RV Reducer with Tooth Profile Modifications 2.1. Contact Force Calculation 2.1.1. Contact Force Model A traditional cycloid drive is the basis used to analyze an RV reducer, and Figure1 shows a traditional cycloid drive coordinate system. Here, xoy and xcoc yc represent the static coordinates of a pin wheel and cycloid gear, O and Oc denote the center of the pin wheel and cycloid gear, and rp and rc denote the pitch circle radius of a pin wheel and cycloid gear, respectively. Pitch point P is the instant velocity center of a cycloid drive and OOc is the position of the crank. When the crank rotates Appl. Sci. 2019, 9, 4099 3 of 19 Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 19 θcr, the cycloid gear will rotate θcy according to the transmission ratio. The rotation of a cycloid gear is θ θ outputrotates using cr , the a parallelogramcycloid gear will mechanism. rotate cy accor Figure 1. Traditional cycloid drive coordinates. Figure 1. Traditional cycloid drive coordinates. The track of the meshing point of a cycloid-pin in a static coordinate system is part of the circular arc ofThe the track pin. According of the meshing to the meshingpoint of principle a cycloid-pin of a cycloid in a static drive, coordinate the tooth profile system of is a cycloidpart of gear the cancircular be obtained arc of the as pin. follows According [20]: to the meshing principle of a cycloid drive, the tooth profile of a cycloid gear can be obtained as follows [20]: 8 i 1 n o H H H H > x = Rp + DRp cos[ 1 i )θ a cos(i θ) + rrp + Drrp S− 2 Kcos(i θ) cos[ 1 i θ] < =( + )[(1− −)]−(i − ) + ( + ) 1(n )− − [ − (1− )]o , (1) > H H H H : y = Rp + DRp sin[ 1 i )θ + a sin(i θ) rrp + Drrp S − 2 Ksin(i θ) + sin[ 1 i θ] , (1) − − − =( +)[(1− )]+( ) − ( +) ( ) + [ (1− )] where where Zp iH = , (2) Zc H Z p i = , (2) aZZpc K = , (3) Rp + DRp aZ p S = 1K+=K2 2Kcos θ, .(3) (4) RR−pp+Δ Here, θ represents the angle of the pin position vector; DRp and Drrp refer to the modification coefficient of the moved distance and equidistance,=1+ respectively;−2.Zc and Zp denote the cycloid disc teeth(4) number and pin number; rrp and Rp represent the pin radius and pin center circle radius, respectively; Here,represents the angle of the pin position vector; ΔRp and Δrrp refer to the modification and a represents the eccentricity. coefficient of the moved distance and equidistance, respectively; and denote the cycloid disc The pitch circle radius of a pin wheel and cycloid gear can be calculated using the following teeth number and pin number; rrp and Rp represent the pin radius and pin center circle radius, equations: respectively; and represents the eccentricity.( rp = aZp The pitch circle radius of a pin wheel and cycloid gear can be calculated using the following(5) rc = aZc equations: Figure2 shows the mechanism of an RV reducer.
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